Is there a more consistently unpredictable figure in modern pop than Bjork? It’s certainly hard to think of another artist who seems so wilfully addicted to perversity, both in her image and her music, and who simultaneously seeks and evades attention to quite the same degree. Ever since she appeared on stage as the heavily pregnant teenage lead singer of an Icelandic punk band, dressed only in a ripped, navel-revealing T-shirt (this was more than a decade before it became a statement of faith for female celebrities to flaunt their pregnancies), she has been a source of perplexed fascination, both at home in Iceland and abroad. A natural exhibitionist, she arrived for last year’s Cannes premiere of Lars von Trier’s Dancer in the Dark (for which her hypnotic performance as an unhappy Czech immigrant in 1950s America won her the best actress award) dressed as a Christmas tree. At this year’s Oscar ceremony, she chose something more sensible—a swan’s costume, which receives another outing on the cover of her new album, Vespertine (which means “relating to the evening”).

But, at times, it has all been too much for Bjork: in 1996, one of her American fans filmed himself in the process of sending her a letter bomb. He then committed suicide by shooting himself in the head. A few months later, at an airport in Thailand, Bjork attacked a journalist who was attempting to interview her son, Sindri, wrestling the startled woman to the ground.

After that, she seemed to withdraw into a thicket of introspection. Here was a woman retreating… what exactly? She left London after the break-up of her relationship with the DJ Goldie, returning to her native Iceland, where she began tentative work on her 1997 album, Homogenic. A collection of loose, impressionistic songs, it is one of the strangest albums in recent pop history. Gone were the punchy dance rhythms that made her first two albums, Debut (1993) and Post( 1995), commercially so successful. In their place was a series of intense, brooding soundscapes, an epic music of loneliness and dislocation, which baffled in all its complex oddity as Bjork sang of healing herself with a razor blade. I thought it was terrific. Most critics disagreed, however, dismissing Homogenic, with its orchestral grandeur and lyrical opacity, as a work of pure self-indulgence—a “cry for help from a woman on the edge”, as one critic put it. On the edge of what, you wondered.

Her new album, Vespertine, is more immediately accessible than Homogenic, but no less impressive. There is nothing mechanical or calculating about this music; each song assumes the form of a complex emotional drama, a stylised confession that, at its best, has the loose, improvised feel of free-form jazz. Bjork, you feel, has no idea what might happen when she enters a recording studio. As a lyricist, she occupies a dream-world, somewhere between innocence and experience. Her songs are often obliquely sexual, emotion recollected not in tranquillity exactly, but in rapture: “He slides inside/Half awake half asleep/We faint back into sleephood/When I wake up the second time in his arms/gorgeousness” (“Cocoon”). And there is an often unremarked humour in her work, particularly in the way she deliberately mangles language, having fun with mispronunciation and neologisms, the way her voice does not always complement the music so much as work antagonistically against it.

Those who know Bjork well, or have worked closely with her, are impressed by something unaccountable in her personality—the unaccountability of true talent. “Bjork is very special,” said Catherine Deneuve, with whom she co-starred in Dancer in the Dark. “She cannot really act, she can only feel.” This is a perceptive remark: “You don’t have to speak/I feel/emotional landscapes/they puzzle me,” sang Bjork on “Joga”, one of the best tracks on Homogenic.

For U2’s Bono, it is less Bjork’s wayward personality and more her music—particularly the “unforgettable power” of her voice—that most compels. Bono is right about that voice, which is a vast, tortured, other-worldly instrument that sounds like nothing else because it is like nothing else. To listen to Bjork singing is often to be returned to the anguish and pain of your first failed love affair: even when she is happy, she sounds sad, as if every word is being delivered in an ecstasy of suffering.

“I never thought I would compromise,” she sings on “Unison”, the closing track on Vespertine. But this is not a work of compromise; it is scorched through with the authenticity of an artist in the process of a complicated journey of self-invention. Where she might end up, or where her talent might take her, is impossible to say. What matters is the journey itself, and it will be worth following her as she matures and deepens and continues to experiment. As for that voice? Well, as Bono said after he listened to Bjork rehearsing one afternoon in a darkened studio: “It’s like an icepick—it goes straight to the heart.”